Equiangular polygon
Encyclopedia
In Euclidean geometry
, an equiangular polygon is a polygon
whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon
.
The only equiangular triangle is the equilateral triangle. Rectangle
s, including the square, are the only equiangular four-sided figures.
For an equiangular n-gon each angle is 180° − 360°/n; this is the equiangular polygon theorem.
Viviani's theorem
holds for equiangular polygons (and also holds for equilateral ones):
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, an equiangular polygon is a polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
.
The only equiangular triangle is the equilateral triangle. Rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
s, including the square, are the only equiangular four-sided figures.
For an equiangular n-gon each angle is 180° − 360°/n; this is the equiangular polygon theorem.
Viviani's theorem
Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from a point to the sides of an equilateral triangle equals the length of the triangle's altitude....
holds for equiangular polygons (and also holds for equilateral ones):
- The sum of distances from a point to the side lines of an equiangular [or equilateral] polygon does not depend on the point and is that polygon's invariant.
External links
- A Property of Equiangular Polygons: What Is It About? a discussion of Viviani's theorem at Cut-the-knotCut-the-knotCut-the-knot is a free, advertisement-funded educational website maintained by Alexander Bogomolny and devoted to popular exposition of many topics in mathematics. The site has won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003,...
.